Part I, Chapter III — Critical & Symbolic Notes¶
Real Cyclotomic Minimal Polynomials, Kepler’s Heptagon, and Fermat Number Status
Exact Polygon Classifier in geometor.klein.constructibility¶
Combining §§ 2, 7, and 8 yields the complete Gauss–Wantzel Theorem: a regular $n$-gon ($n \ge 3$) is constructible with straightedge and compasses if and only if Euler’s totient $\varphi(n) = \deg \Phi_n(z)$ is a power of $2$:
where $a \ge 0$ and $p_1 < p_2 < \dots < p_r$ are distinct Fermat primes ($3, 5, 17, 257, 65537, \dots$).
In geometor.klein.constructibility.cyclotomic_real_minimal_poly(), we also compute the exact minimal polynomial $\psi_n(u) \in \mathbb{Q}[u]$ of the real chord coordinate $u = z + z^{-1} = 2\cos\!\left(\frac{2\pi}{n}\right)$ of degree $\frac{1}{2}\varphi(n)$:
$n$ |
$\varphi(n)$ |
Cyclotomic $\Phi_n(z)$ |
Real Minimal Polynomial $\psi_n(u)$ ($u = 2\cos\frac{2\pi}{n}$) |
|---|---|---|---|
$7$ |
$6$ |
$z^6 + z^5 + z^4 + z^3 + z^2 + z + 1$ |
$u^3 + u^2 - 2u - 1$ (deg $3$) |
$9$ |
$6$ |
$z^6 + z^3 + 1$ |
$u^3 - 3u + 1$ (deg $3$) |
$11$ |
$10$ |
$z^{10} + \dots + z + 1$ |
$u^5 + u^4 - 4u^3 - 3u^2 + 3u + 1$ (deg $5$) |
$13$ |
$12$ |
$z^{12} + \dots + z + 1$ |
$u^6 + u^5 - 5u^4 - 4u^3 + 6u^2 + 3u - 1$ (deg $6$) |
Historical & Sister-Corpus Notes¶
Kepler’s *Inscibilis* Heptagon (`geometor/kepler`): Nearly two centuries before Gauss and Wantzel, Johannes Kepler argued in Harmonices Mundi, Liber I, Props. XLV–XLVII (1619) that the regular heptagon ($n = 7$) is inscibilis (“unknowable” by plane geometry) because its chord determination leads to an irreducible solid/cubic problem ($u^3 + u^2 - 2u - 1 = 0$) possessing no prior ground in planar bisections.
Status of $F_\mu = 2^{2^\mu} + 1$ After 1895: In § 1, Klein notes that in 1895 $F_5, F_6, F_7$ were known to be composite while $F_8$ remained open ($F_7$ and $F_8$ were proved composite by Morehead and Western in 1905 and 1909; today $F_5$ through $F_{32}$ are all known to be composite, leaving Klein’s prophetic remark that $\mu = 4$ may be the last prime case completely unrefuted!).